Snap off and discard woody bases from asparagus. Trim stalks barely longer than the bread slices. Place a steamer rack in the bottom of a large saucepan and add water to just below the basket. Bring to boiling. Add asparagus. Cover and reduce heat. Steam for 3 to 5 minutes or until crisp-tender. Transfer asparagus to a bowl of ice water for 30 seconds to cool; drain.
Using kitchen shears, cut each piece of prosciutto in half lengthwise. Starting at the base of an asparagus spear, wrap 1 halved prosciutto piece around the asparagus spear diagonally to the top of the spear. Repeat with remaining asparagus and prosciutto.
Spread four slices of the bread with mustard. Place four prosciutto-wrapped asparagus spears horizontally on each mustard-topped bread slice. If desired, top with arugula. Add two slices of cheese to each sandwich, arranging slices to completely cover bread. Brush the remaining four bread slices lightly with balsamic vinaigrette. Place bread, vinaigrette sides down, on top of cheese.
Preheat a panini grill according to manufacturer's directions (or see tip, page xx); place sandwiches (half at a time, if necessary) in grill. Close lid; cook for 4 to 5 minutes or until cheese melts.